Question Number 173228 by mathlove last updated on 08/Jul/22
Answered by Frix last updated on 08/Jul/22
$$\mathrm{let}\:\mathrm{sin}\:\frac{\pi}{\mathrm{11}}\:={s}\:\wedge\:\mathrm{cos}\:\frac{\pi}{\mathrm{11}}\:={c} \\ $$$$\left(\mathrm{1}\right)\:\Rightarrow\:{y}={s}^{\mathrm{2}} {x}−\frac{{s}^{\mathrm{4}} }{\mathrm{4}} \\ $$$$\left(\mathrm{2}\right)\:\Rightarrow\:{y}=\frac{{x}}{{c}^{\mathrm{2}} }+\frac{{c}^{\mathrm{2}} }{\mathrm{4}} \\ $$$$\Rightarrow \\ $$$${x}=−\frac{{c}^{\mathrm{2}} }{\mathrm{4}}\wedge{y}=−\frac{{s}^{\mathrm{2}} }{\mathrm{4}} \\ $$$$\Rightarrow\:\sqrt{−{x}}+\sqrt{{y}}=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{e}^{\mathrm{i}\frac{\pi}{\mathrm{11}}} \wedge\left(\sqrt{−{x}}+\sqrt{{y}}\right)^{\mathrm{2024}} =\mathrm{2}^{\mathrm{2024}} \\ $$
Commented by mathlove last updated on 09/Jul/22
$${thanks} \\ $$